Covalent organic frameworks (COF) are deemed as disruptive membrane materials owing to their versatile functionalities, high stability and ordered nanochannels. However, the size mismatch between the COF intrinsic pore and gas molecules becomes a grand challenge when using COF membrane for carbon capture. Herein, we propose a new type of polymer-functionalized COF laminar membranes through grafting CO2-phlic polyethyleneimine (PEI) polymer onto carboxylic acid COF (cCOF) laminar membranes. The cCOF nanosheets with high density of carboxyl groups are prepared via modified single phase solution method and then assembled to fabricate membranes. Further, the cCOF membranes are functionalized with PEI using 1-ethyl-3-(3-dimethylaminopropyl) carbodiimide/N-Hydroxysuccinimide (EDC/NHS) coupling chemistry. The introduction of abundant amino groups from PEI afford the affinity of membranes toward CO2. Meanwhile, the graft of the PEI polymers reduces the defects or pinholes on the membrane surface. As a result, the PEI functionalized cCOF membranes exhibit a high CO2 permeance of 1004 GPU and a CO2/N2 selectivity of 33.7, which reach the carbon capture target performance and show great prospect for treating real flue gas. Our approach of functionalizing cCOF membranes with tailored functionalities may inspire the design of molecule-selective COF membranes.